Original mathematical art generated by parametric curves, polar roses, fractals, and chaotic attractors — inspired by the beauty of equations, not copied from any artist.
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Pythagorean Theorem Right triangles traced as perpendicular Lissajous oscillations.
a 2 + b 2 = c 2 a^2 + b^2 = c^2 a 2 + b 2 = c 2 Eulers Identity Rotating epicycles summing to a circle — the geometry behind e^(iθ).
e i π + 1 = 0 e^{i\pi} + 1 = 0 e iπ + 1 = 0 Taylor Series Stacked harmonics approximate smooth curves — Taylor's idea made visible.
f ( x ) = ∑ n = 0 ∞ f ( n ) ( a ) n ! ( x − a ) n f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n f ( x ) = ∑ n = 0 ∞ n ! f ( n ) ( a ) ( x − a ) n Fourier Transform Epicycles decompose complex signals into pure frequencies.
X ( f ) = ∫ x ( t ) e − i 2 π f t d t X(f)=\int x(t)e^{-i2\pi ft}\,dt X ( f ) = ∫ x ( t ) e − i 2 π f t d t Bayes Theorem A belief network — evidence rewiring connections between hypotheses.
P ( A ∣ B ) = P ( B ∣ A ) P ( A ) P ( B ) P(A|B)=\frac{P(B|A)P(A)}{P(B)} P ( A ∣ B ) = P ( B ) P ( B ∣ A ) P ( A ) Newtons Second Law Nested orbits — force and acceleration in circular motion.
Universal Gravitation Logarithmic spirals model gravitational clustering in galaxies.
F = G m 1 m 2 r 2 F = G\frac{m_1 m_2}{r^2} F = G r 2 m 1 m 2 Bernoulli Equation Superposed waves echo pressure and velocity trade-offs in flowing fluids.
P + 1 2 ρ v 2 + ρ g h = const P + \frac{1}{2}\rho v^2 + \rho gh = \text{const} P + 2 1 ρ v 2 + ρ g h = const Navier Stokes Equation Turbulent wave interference — viscosity smoothing chaotic flow.
ρ ( ∂ t v + v ⋅ ∇ v ) = − ∇ P + μ ∇ 2 v \rho(\partial_t \mathbf{v} + \mathbf{v}\cdot\nabla\mathbf{v}) = -\nabla P + \mu\nabla^2\mathbf{v} ρ ( ∂ t v + v ⋅ ∇ v ) = − ∇ P + μ ∇ 2 v Wave Equation Traveling sine superpositions — the mathematics of sound and light.
∂ 2 u ∂ t 2 = c 2 ∇ 2 u \frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u ∂ t 2 ∂ 2 u = c 2 ∇ 2 u Heat Equation Diffusing waves — heat spreading from hot to cold regions.
∂ u ∂ t = α ∇ 2 u \frac{\partial u}{\partial t} = \alpha \nabla^2 u ∂ t ∂ u = α ∇ 2 u Ideal Gas Law A polar rose — pressure and volume oscillating in thermal equilibrium.
Entropy Equation The Lorenz attractor — deterministic chaos mirroring irreversible disorder.
S = k B ln Ω S = k_B \ln \Omega S = k B ln Ω Maxwell Equations Oscillating fields drawn as Fourier epicycles — light as coupled E and B waves.
∇ ⋅ E = ρ ε 0 \nabla\cdot\mathbf{E} = \frac{\rho}{\varepsilon_0} ∇ ⋅ E = ε 0 ρ Lorentz Force Charged particles curving in combined electric and magnetic fields.
F = q ( E + v × B ) \mathbf{F} = q(\mathbf{E} + \mathbf{v}\times\mathbf{B}) F = q ( E + v × B ) Einstein Mass Energy Golden-angle seed packing — nature's efficiency echoing concentrated energy.
Escape-Time Fractals Premium
Schrodinger Equation Julia set boundaries — infinite complexity from a simple quadratic map.
i ℏ ∂ ψ ∂ t = H ^ ψ i\hbar\frac{\partial\psi}{\partial t} = \hat{H}\psi i ℏ ∂ t ∂ ψ = H ^ ψ Escape-Time Fractals Premium
Heisenberg Uncertainty Fractal coastlines — zoom forever and detail never resolves, like uncertainty.
Δ x Δ p ≥ ℏ 2 \Delta x \Delta p \geq \frac{\hbar}{2} Δ x Δ p ≥ 2 ℏ Tsiolkovsky Rocket Equation Staging as nested orbits — each burn shedding mass to climb higher.
Δ v = v e ln m 0 m f \Delta v = v_e \ln\frac{m_0}{m_f} Δ v = v e ln m f m 0 Shannon Entropy Information networks — surprise distributed across connected nodes.
H = − ∑ i p i log p i H = -\sum_i p_i \log p_i H = − ∑ i p i log p i Sir Model Epidemic waves — susceptible, infected, and recovered oscillating through time.
d S d t = − β S I , d I d t = β S I − γ I \frac{dS}{dt}=-\beta SI,\; \frac{dI}{dt}=\beta SI-\gamma I d t d S = − β S I , d t d I = β S I − γ I Logistic Growth Saturating growth unfolding as a four-petal rose — capacity as geometry.
d N d t = r N ( 1 − N K ) \frac{dN}{dt}=rN\left(1-\frac{N}{K}\right) d t d N = r N ( 1 − K N ) Hubble Law Expanding spiral arms — recession velocity growing with cosmic distance.
Gradient Descent A loss landscape as a web — each step sliding downhill along connections.
θ t + 1 = θ t − η ∇ L ( θ t ) \theta_{t+1}=\theta_t-\eta\nabla L(\theta_t) θ t + 1 = θ t − η ∇ L ( θ t ) Transformer Attention Dense attention weights as a fully connected constellation of tokens.
Attention ( Q , K , V ) = softmax ( Q K T d k ) V \text{Attention}(Q,K,V)=\text{softmax}\!\left(\frac{QK^T}{\sqrt{d_k}}\right)V Attention ( Q , K , V ) = softmax ( d k Q K T ) V